Vertical Point Sampling with a Camera

Vertical point sampling has seen relatively little use in practical forestry, in part because existing field techniques are difficult. We show how vertical point sampling can be implemented quickly and easily using a camera. We give tables and equations for calculating the height-squared factor, which plays a role similar to that of the basal area factor in horizontal point sampling. Some suggestions for choosing a height-squared factor are discussed, along with potential applications for further exploration. We illustrate the technique using a case study in southern Maine. Direct estimates with no statistically detectable bias were obtained using height-squared factors greater than 3. The results also suggested that the technique could be used as a correlate in double sampling for variables such as cubic volume, stand density index, and biomass, and possibly board foot volume as well.

Economic Impacts of Projecting Horizontal Angles to the Wrong Height When Conducting Point Sampling

Horizontal point sampling selects sample trees by projecting horizontal angles. In many inventories, angles are to be projected to dbh, or the diameter at 4.5 ft, but because of user error, angles are often projected to heights other than breast height. Thus, errors are made as to which trees should be sampled, probabilities of sampling individual trees are incorrect, and the basal area estimate does not truly correspond to dbh. The objective of this study was to determine the potential economic impacts of projecting angles at heights other than breast height when breast height is the desired height. Projections for two planting densities (400 and 1,000 seedlings per acre) and two ages (20 and 30) were used to establish virtual plantations, and sampling was conducted using 10 and 20 basal area factor prisms by projecting horizontal angles to four heights, 4.5, 5.0, 5.5, and 6.0 ft. A taper equation was used to estimate changes in diameter along the stem. For the stand conditions examined in this study, incorrectly projecting angles to heights other than breast height reduced timber appraisals by as much as $190/ac. Across many acres and stands, this type of nonsampling error can result in serious errors in valuing stumpage.

A note on the relationship between the quadratic mean stand diameter and harmonic mean basal area under size-biased distribution theory

This note seeks to extend the utility of size-biased distribution theory as applied to forestry through two relationships regarding the quadratic mean stand diameter. First, the quadratic mean stand diameter's relationship to the harmonic mean basal area for horizontal point sampling, which has been known algebraically from early on, is proved under size-biased distribution theory. Second, a new result, which may prove most valuable in viewing the graphical representation of assumed distributions, is also derived. The results are also shown to apply to the basal area – size distribution, providing a unique duality between the two means.

Some observations on fitting assumed diameter distributions to horizontal point sampling data

This paper revisits the link between assumed diameter distributions arising from horizontal point samples and their unbiased stand-based representation through weighted distribution theory. Examples are presented, which show that the assumption of a common shared parameter set between these two distributional forms, while theoretically valid, may not be reasonable in many operational cases. Simulation results are presented, which relate the conformity (or lack thereof) in these estimates to sampling intensity per point and the underlying shape of the population diameter distribution from which the sample point was drawn. In general, larger sample sizes per point are required to yield reliable parameter estimates than are generally taken for inventory purposes. In addition, a complimentary finding suggests that the more positively skewed the underlying distribution, the more trees per point are required for good parameter estimates.

Minimizing the Rounding Error from Point Sample Estimates of Tree Frequencies

Three solutions are presented for estimating stems per acre when trees are tallied by diameter class with horizontal point sampling. The first solution is based on the arithmetic mean of the diameter-class limits. The second is based on the geometric mean of the diameter-class limits and is unbiased for uniform within-class diameter distributions. The third is a harmonic mean solution; it is derived from the ratio of the geometric mean squared and the arithmetic mean. If the within-class distribution is linear, then the solution based on the geometric mean is preferable. Any of these solutions may be adequate provided diameter-class widths are minimized, particularly for small trees. The geometric-mean solution is the recommended solution in instances where uniformity can be assumed within diameter classes. Observed frequencies in two stands, however, provided inconsistent results. In estimating tree frequency, errors resulting from grouping become smaller as tree diameter increases. For large trees, estimation of frequency can be obtained with much wider diameter limits than with small trees for a fixed level of grouping error. Rather than fixing diameter-class width, it may be advantageous to consider a narrower class width or exact measurement for trees in smaller diameter classes. West. J. Appl. For. 12(4):108-114.

Large-scale 35-mm aerial photographs for assessment of vegetation-management research plots in eastern Canada

Two 35-mm cameras were mounted on a boom and suspended from a tethered helium-filled blimp to obtain nominally vertical aerial photographs (1:828 and 1:414 contact scale) of vegetation-management research plots. Photo and ground estimates of woody plant crown area (m2/ha) and rootstock density (number/ha) were compared for several experimental vegetation-control treatments. Horizontal point-sampling estimates of total crown area made directly on 1:93 scale prints (enlarged from 1:414) correlated strongly with equivalent estimates made on the ground (n = 62, r2 = 0.97). An estimated 20 ground-truth plots were required to adequately quantify photo bias and correct subsequent prediction of actual total crown area on the plots studied. Much of the observed photo bias could be attributed to the undersampling of small rootstocks. Exclusion of individual rootstocks less than 0.10 m2 in crown area (or, equivalently, <56 cm in height) resulted in a 1:1 relation between the two sampling methods for estimates of both total crown area (r2 = 0.98) and rootstock density (r2 = 0.97). If data for rootstocks in smaller size classes are not needed, uncorrected photo estimates may be appropriate for evaluation of treatment response. Ground-sampling costs averaged $200 (Canadian) per plot, compared with photo costs of $104 per plot (without ground truth) or $150 per plot (with 20 ground-truth plots). Smaller scale photos (1:828 contact) cost 11% less than the larger scale tested, but resulted in significant undersampling of individual rootstocks less than 0.2 m2 in crown area (or, equivalently, <80 cm in height).

A two-stage method for horizontal point sampling in young forest stands

Horizontal point sampling is sometimes difficult to use in dense stands of small-diameter stems because of poor visibility. One solution to this problem is to use vertical point sampling in the field to obtain a larger than necessary preliminary sample. The diameter and distance of each tree in the vertical point sample is measured, and then a computational procedure identical with horizontal point sampling is used to subsample the vertical point sample tree list. For the method to work, it is necessary that the horizontal point sampling criterion be at least as limiting in a certain sense as the vertical point sampling criterion. Our principal result is to show how this necessary condition can be met.

Combining taper functions and critical height sampling for unbiased stand volume estimation

New estimators have been developed for critical height sampling that allow any taper function to be used as a variance reduction mechanism. The new estimators are compatible with current horizontal point sampling procedures and will lead to unbiased estimates of cubic volume and cubic volume growth. It is proposed that critical height sampling should replace horizontal point sampling when one upper stem measurement can be made, because horizontal point sampling relies on predicted tree volumes and the assumption that resulting stand volume predictions are unbiased.

Aspects of statistical bias due to the forest edge: horizontal point sampling

A tree-concentric procedure is presented to eliminate edge effect statistical bias for horizontal point sampling. For known forest populations, the exact mean of the unbiased (adjusted for edge effect) and biased (unadjusted for edge effect) estimators of a forest characteristic can be determined along with the exact bias. Edge-effect bias is investigated for three forests that vary in area, four basal area factors (BAF), and three forest characteristics. Exact and estimated results based on 5000 random points were compared. Edge effect bias increases as the BAF decreases and varies with forest size, size and spatial distribution of trees, percentage of edge trees, and forest characteristic. The variance of the biased estimator was always smaller than the variance of the unbiased estimator. Using mean square errors, the biased estimator was found to be, in general, more accurate and the distortion of probability statements caused by the bias negligible for small to moderate sample sizes, especially for larger BAF's and certain forest characteristics.